MinT - Best Known (39−21, 39, s)-Nets in Base 2

Best Known (39−21, 39, s)-Nets in Base 2

(39−21, 39, 17)-Net over F₂ — Constructive and digital

Digital (18, 39, 17)-net over F₂, using

t-expansion [i] based on digital (15, 39, 17)-net over F₂, using
- net from sequence [i] based on digital (15, 16)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(39−21, 39, 18)-Net over F₂ — Digital

Digital (18, 39, 18)-net over F₂, using

net from sequence [i] based on digital (18, 17)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 18 and N(F) ≥ 18, using
  - a curve from the manYPoints database [i]

(39−21, 39, 44)-Net over F₂ — Upper bound on s (digital)

There is no digital (18, 39, 45)-net over F₂, because

1 times m-reduction [i] would yield digital (18, 38, 45)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2³⁸, 45, F₂, 20) (dual of [45, 7, 21]-code), but
  - residual code [i] would yield linear OA(2¹⁸, 24, F₂, 10) (dual of [24, 6, 11]-code), but
    - residual code [i] would yield linear OA(2⁸, 13, F₂, 5) (dual of [13, 5, 6]-code), but
      - 1 times truncation [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
        bound by Fontaine and Peterson [i]

(39−21, 39, 46)-Net in Base 2 — Upper bound on s

There is no (18, 39, 47)-net in base 2, because

1 times m-reduction [i] would yield (18, 38, 47)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2³⁸, 47, S₂, 20), but
  - the linear programming bound shows that MÂ â‰¥ 140 737488 355328 / 385 > 2³⁸ [i]